j. ambjørn et al- lorentzian 3d gravity with wormholes via matrix models

8/3/2019 J. Ambjørn et al- Lorentzian 3d gravity with wormholes via matrix models

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1 Introduction

A major task of modern theoretical physics is to unite quantum mechanics with thetheory of gravity and to understand “quantum geometry”. In four dimensions, this isproving a difficult task and there is no general consensus on which direction to take.

The so-called “string community”, originating from quantum field theory, claimsthat M-theory provides the only viable road to unifying quantum mechanics andgravity, while people coming from the theory of relativity tend to favour approachesbased on canonical quantization. So far neither of these approaches has given us adetailed understanding of the microscopic quantum geometry of the real space-time.

The situation is considerably better in dimension d < 4. Although lower-dimensional models do not possess propagating gravitational degrees of freedom,their geometries are still subject to quantum fluctuations, and the quantum the-ories are non-trivial. One may use conventional quantum field-theoretic methodsto investigate the coupling between matter and gravity, and to define and calcu-late diffeomorphism-invariant correlation functions and the dynamically generated

fractal dimension of quantum space-time. Of course, we have no way of knowinghow relevant the study of the lower-dimensional theories will be for the eventualtheory of four-dimensional quantum gravity, but it is clear that there are structuralsimilarities and certainly some of the same questions can be asked.

A detailed and explicit analysis exists in two space-time dimensions. One cancalculate the anomalous dimensions acquired by matter fields when coupled to 2dEuclidean gravity [1, 2]. It is also understood that a typical two-dimensional Eu-clidean geometry contributing to the gravitational path integral has a fractal struc-ture with Hausdorff dimension four, much in the same way as a typical path in thepath integral for a particle is also fractal, with Hausdorff dimension two.

The fractal structure is best understood by introducing a geodesic “time” on thetwo-geometries. Euclidean 2d gravity is characterized by the fact that an infinitenumber of baby universes branches off the one-dimensional spatial slice as it evolvesin this “time” [3, 4, 5]. While such a process is unavoidable in two-dimensionalgravity models coming from string theories (describing non-critical strings), thereis nothing in a theory of quantum gravity which demands that space should beallowed to split into disconnected parts. In the context of canonical gravity thiswould require a “third quantization” to enable the destruction and creation of babyuniverses, a possibility that is not usually considered.

Not allowing for the creation of baby universes leads to a new two-dimensionalquantum theory, called Lorentzian quantum gravity . The name derives from the fact

that the sum over geometries in the path integral includes only a particular subclassof Euclidean geometries which are obtained through a Wick rotation from a set of Lorentzian space-times with a well-defined causal and globally hyperbolic structure[6]. Interestingly, the resulting quantum theory is different from the Euclidean one.For instance, the fractal dimension of a typical geometry is two and not four, andthe coupling of matter and geometry creates no anomalous scaling dimensions for

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the matter fields [7, 8]. The detailed relation between the geometries of the twomodels is well understood [9].

Very important in the study of two-dimensional quantum gravity have beenmethods and concepts from statistical physics [10, 5]. The regularization in boththe Euclidean and Lorentzian case uses so-called dynamical triangulations. In this

approach, geometries are created by gluing together large numbers of identical trian-gular building blocks. The geodesic edge length of the triangles is a measure of thefineness of the simplicial lattice and defines a diffeomorphism-invariant cutoff of thetheory. The action of such a piecewise linear two-geometry is calculated by Regge’sprescription, and the state sum over geometries can in many cases be performedexplicitly. The scaling limit (i.e. taking the number of triangles to infinity), definesthe continuum limit of these models.

Since in both Euclidean and Lorentzian 2d dynamical triangulations we have aconcept of time (“geodesic” time and “proper” time respectively), a transfer matrixcan be introduced which describes the evolution (the transition amplitude) betweenthe spatial configurations at time t and t + a, where a is a discrete lattice spacing.

The notion of a transfer matrix is familiar from quantum field theories on fixedlattices. It allows us to extract the continuum Hamiltonian of the system in thelimit as a → 0 according to

φ(x)|T |φ′(x′) = φ(x)|e−aH |φ′(x′) → φ(x)|(1 − aH + O(a2))|φ′(x′). (1)

This strategy has been applied successfully in Euclidean and Lorentzian two-dimen-sional quantum gravity.

While all of this works beautifully in dimension two, where it has given us a num-ber of powerful analytical tools, the situation is quite different when one tries to usethe method of dynamical triangulations to obtain a theory of Euclidean quantumgravity in higher dimensions. Such models have been investigated mainly throughnumerical simulations [11, 12], and the results have so far been disappointing: nointeresting continuum limits seem to exist [13].2 In fact, this was one of the mainmotivations for constructing alternative Lorentzian models of dynamical triangula-tions, which we have already mentioned in the two-dimensional context. They haverecently been shown to exist as well-defined regularized models of quantum gravityalso in three and four space-time dimensions [15, 16] (see also [17] for a review of discrete Lorentzian gravity).

An investigation of the continuum properties of the three-dimensional Lorentziangravity model has already begun. Computer simulations show that it avoids some

of the problems of Euclidean simplicial quantum gravity and most likely has a con-tinuum limit [18], thus fulfilling some of the hopes raised by the two-dimensionalLorentzian model. As is well-known from previous attempts, analytic tools are hardto come by in statistical models of quantum geometry in d > 2. In particular,

2This seems to be part of a general pattern, since also other non-perturbative discrete approachesto four-dimensional Euclidean quantum gravity have had little success, see [14] for a review.

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the matrix-model methods that proved so powerful in two dimensions have not yetbeen made into a useful calculational tool in higher-dimensional quantum gravity.The purpose of this article is to demonstrate that in three-dimensional Lorentzianquantum gravity, such analytic matrix-model techniques can indeed be employed.

The remainder of this article is organized as follows. In Sec. 2 we describe

the quantum gravity model in terms of simplicial geometries and define its trans-fer matrix. At the discretized level, it is a simple variant of the three-dimensionalLorentzian model introduced in [15, 16] and studied by Monte Carlo simulationsin [18, 19]. Motivated by some well-known properties of (2+1)-dimensional quan-tum gravity, we perform an integration over all but one of the spatial geometricdegrees of freedom in Sec. 3. In the following Sec. 4 we remind the reader of howto obtain the continuum Hamiltonian from the resulting transfer matrix. The cor-respondence of 3d Lorentzian quantum gravity with the already partially solvedtwo-matrix model [20] with ABAB-interaction is established in Sec. 2. We rein-terpret the phase structure of this matrix model in terms of geometry in Sec. 6.This also involves a discussion of other, closely related matrix models with so-called

touching-interactions. We briefly comment on the status of the full and as yetunsolved ABAB-matrix model in Sec. 7. In Sec. 8 we describe how taking the con-tinuum limit in the ABAB-model fits in with our previous considerations of thislimit in three-dimensional gravity. We end with an outlook in Sec. 9. The appendixcontains a derivation of the gravitational action in terms of the 3d building blocksused in this article.

2 Lorentzian 3d gravity from pyramids and tetra-

hedra

The motivation for constructing non-perturbative gravitational path integrals of Lorentzian geometries and the general properties of the dynamically triangulatedmodel in three dimensions were described in [15, 16]. For our present purposes itis convenient to consider a slightly modified regularization which can be related toa quartic matrix model. (The cubic matrix model that would correspond to ouroriginal model which uses only tetrahedra has not yet been solved.)

In the regularized model, (proper) time t is discretized into integer lattice steps of unit one. The spatial slices at t = 0, 1, . . . are piecewise linear manifolds of sphericaltopology, constructed by gluing together flat squares with edge length ls = a (ratherthan the equilateral triangles with edge length ls = a of references [15, 16, 18]).

The geometry of a spatial slice is uniquely fixed by this length assignment and byits connectivity matrix, specifying which pairs of squares are glued together alonga common edge. Two spatial slices at t and t+1 form the space-like boundaries of a three-dimensional piecewise-linear manifold “sandwich” that lies in between theslices and has topology S 2 × [0, 1]. The fundamental three-dimensional buildingblocks used for “filling in” are regular pyramids with square base and tetrahedra

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(4,1)

(1,4) (2,2)

t+1

t

Figure 1: Pyramids and tetrahedra can be used to discretize 3d Lorentzian space-times. We show the three types of fundamental building blocks and their locationwith respect to the spatial hypersurfaces of constant integer-t.

(see Fig. 1). The pyramids have either their base in the t-plane and their tip in

the t+1-plane – in which case we call them (4,1)-pyramids – or vice versa for the(1,4)-pyramids. In addition, each pyramid has four time-like edges of equal squaredlength l2

t connecting the neighbouring slices3. Two pyramids of equal orientationcan be glued together along a time-like triangle. Since (4,1)-pyramids with base att and (1,4)-pyramids with base at t + 1 cannot share a triangular face, we need anadditional type of building block, namely, a (2,2)-tetrahedron with one space-likeedge (of length ls) each in the t- and t+1-planes (Fig. 1). Its remaining four time-likeedges have again squared length l2

t , so that its faces can be glued to the pyramidsof both types.

We will in the following use the term “quadrangulation” to denote a piecewise-flat geometry made of (4,1)-, (1,4)- and (2,2)-building blocks, as well as their two-dimensional spatial sections. (Note that a 3d quadrangulation may be thought of as a particular kind of a three-dimensional tri angulation, obtained by cutting allpyramids into pairs of tetrahedra.) Starting from a S 2-quadrangulation at timet = 0, we can by successive gluing build up a three-dimensional space-time of lengtht in the time direction, consisting of t +1 spatial spheres, and t “sandwiches” inbetween.

Although our choice of allowed discretized space-times is of course motivatedby the causal structure associated with the physical Lorentzian signature, we willfrom now on do all calculations for the already Wick-rotated Euclidean geometries.Without loss of generality, we set l2

t = βl2s ≡ βa2, for β > 1/2 (because of triangle

inequalities). Next, we must compute the Boltzmann weight e−S

associated with agiven three-geometry. The Euclidean Einstein action in the continuum is given by

S cont = − 1

16πGN

M

d3x

g(x) (R(x) − 2Λ) − 1

8πGN

∂M

d2x

h(x) K (x), (2)

3Our language here is “Lorentzian” in line with our general philosophy [ 15, 16], where a Wickrotation corresponds to a sign flip of l2t .

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where h(x) is the induced metric and K (x) the trace of the extrinsic curvature onthe boundary ∂M of the manifold M , and GN and Λ are the gravitational andcosmological coupling constants. As usual for 3d piecewise linear manifolds, thecurvature is concentrated on the one-dimensional edges or links, and proportionalto the deficit angle under rotation around a link in a plane perpendicular to it.

Using the standard Regge prescription for computing the total scalar curvature of asimplicial manifold, the discrete counterpart of (2) is derived in the appendix, wherewe also discuss the inverse Wick rotation of the action.

Let us adopt the following notation: spatial quadrangulations at integer-t arecalled T (t) and a sandwich geometry G(T (t), T (t + 1)). One piece of informationcontained in the data characterizing the geometry G(T (t), T (t+1)) in the interval[t, t+1] is the set of three numbers N 41(t), N 14(t) and N 22(t) of building blocks of the three types. For a given geometry G, the action depends only on these threebulk variables,

S [G

(T

(t),T

(t+1))] = c0

−kN 41(t)+N 14(t)

−N 22(t)+ λN 41(t)+N 14(t)+

1

2N 22(t).

(3)In (3), N 41(t) + N 14(t) −N 22(t) is proportional to the integrated scalar curvature(including the extrinsic curvature terms for the boundaries) between t and t+1, whileN 41(t)+N 14(t)+ 1

2 N 22(t) is proportional to the three-volume of this piece of space-

time. The dimensionless coupling constants k and λ are proportional to the bareinverse gravitational coupling constant 1/GN and the bare cosmological couplingconstant Λ. The explicit form of the coupling constants as functions of β , GN andΛ can be found in the appendix.

When we start stacking up sandwich geometries, the action (3) is by constructionadditive, so that the total action for a space-time extending t steps in the time-

direction becomes

S [k, λ] = c0t − k(N 41 + N 14 − N 22) + λ(N 41 + N 14 +1

2N 22), (4)

where N 41, N 14 and N 22 denote now the total numbers of (4,1)-, (1,4)- and (2,2)-building blocks.

We can now define the transfer matrix associated with a unit step of proper timeas4

T (t+1)|T |T (t) =

G(T (t),T (t+1))

1

C Ge−S [G(T (t),T (t+1))], (5)

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Following [16], we introduce quantum states |T at fixed t, labelled by inequivalent spatial geometries and normalized according to

T 1|T 2 =1

C T 1

δT 1,T 2 ,T

C T |TT| = 1,

where C T is the symmetry factor of the quadrangulation T .

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where the summation is over all distinct three-geometries G(T (t), T (t+1)) whoseboundary geometries are T (t) and T (t+1). C G denotes the symmetry factor of thequadrangulation G, i.e. the order of the automorphism group of G.

Let T 1 = T (t) contain N 41(T (t)) squares and T 2 = T (t+1) contain N 14(T (t+1))squares. Substituting the sandwich action (3) into (5), we can write

T 2|T |T 1 = e−c0 e−(λ−k)(N 41(T 1)+N 14(T 2))

N max22

N 22=N min22

N (T 1, T 2, N 22) e−( 12λ+k)N 22, (6)

where the sum is over all values N 22 which can occur in sandwich geometries withboundary T 1 ∪ T 2. The combinatorial factor N (T 1, T 2, N 22) counts the number of distinct three-geometries (including the symmetry factor weights) in [t, t+1] for afixed number of (2,2)-tetrahedra.

The Euclideanized amplitude for propagating a spatial geometry T 1 = T (0) atproper time 0 to a later geometry T 2 = T (t) at proper time t is obtained by a t-fold

iteration of the transfer matrix,

G(T 1, T 2, t) = T 2|T t|T 1, (7)

and satisfies the completeness relation

T 2|T t|T 1 =T

T 2|T t1|T C T T |T t2|T 1, (8)

for any split t = t1 +t2 of the total time interval.

3 Integrating out geometriesLet us first recall the situation in classical 3d gravity, on space-times with compactspatial slices Σ(g) of genus g. The degrees of freedom of the theory associated withany spatial slice are the geometries (i.e. the spatial metrics gij modulo spatial dif-feomorphisms), the elements of superspace. A priori , they are genuine field degreesof freedom: for example, each metric can be decomposed uniquely (up to a diffeo-morphism) into a constant-curvature metric gij and a conformal factor according togij(x) = e2λ(x)gij(x). However, a canonical analysis reveals that the conformal factorλ(x) is not a dynamical field degree of freedom, but is completely determined bysolving the constraints. What remains is a finite number of so-called Teichmuller

parameters (none for g = 0, 2 for g = 1 and 6g − 6 for g > 1), coordinatizing thespace of constant-curvature metrics for a given compact spatial manifold of genusg. This implies that for the spherical case with Σ(0) = S 2 not even a finite numberof (classical) degrees of freedom is left after the conformal factor has been fixed5.

5We mean here “bulk” degrees of freedom; depending on the choice of boundary conditions onthe initial and final spatial slices, surface degrees of freedom may be “liberated”.

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This raises the question of how these classical properties are reflected in a gra-vitational path-integral approach of the type we are considering. At the discretizedlevel, we can in principle compute the quantum amplitude between two arbitraryspatial geometries (either at a fixed proper-time distance t, or for an arbitrary dis-tance, obtained by summing the discrete propagator (7) over all positive integers t).

In general, this amplitude will not vanish, since there are always many 3d quadran-gulations interpolating between two given boundaries T 1 and T 2. It implies that ageneric path (i.e. a three-geometry) contributing to the path integral does not obeythe classical constraints, which is not particularly surprising. However, what one isreally interested in is the behaviour of these amplitudes in the continuum limit.

Previous investigations of the non-perturbative gravitational path integral6 aresuggestive of what may be happening in three dimensions. The continuum analysis of [23] found that – subject to a number of plausible conditions – the kinetic term of theconformal factor λ is cancelled in the non-perturbative path integral by a Faddeev-Popov term in the measure. This means that there is no conformal kinetic termin the effective action, and therefore that λ(x) is not a propagating field degree of

freedom. This scenario is corroborated by our numerical simulations of 3d Lorentziangravity [18, 19] which did not show any evidence of the conformal divergence closelyassociated with the kinetic term for the conformal factor.

Motivated by these considerations, we will introduce a vastly reduced set of (Hilbert space) states, by summing over all geometric degrees of freedom of a spa-tial slice of a given two-volume N . Writing |T N for a state corresponding to aquadrangulation with N squares, we define for a fixed spatial topology the state

|N :=1

T N C T N

T N

C T N |T N . (9)

The normalization factor in front of the sum ensures the orthonormality of the“area states”, N |N ′ = δN,N ′. The discrete area N may be thought of as a globalconformal degree of freedom. We do not integrate over N in order to keep controlover the continuum limit and to be able to compare our results with those of similarcontinuum approaches (see, for example, [24, 25]). Using the new states (9), we nowmake the conjecture that

N 14|T |T N 41 − N 14|T |T ′N 41 → 0 for N 14, N 41 → ∞. (10)

Expressed in words, this means that for large areas N 41 and N 14 the expectation value

N 14|T |T N 41 does not depend on which “representative” T N 41 is chosen from the setof 2d quadrangulations with N 41 squares. The conjecture should be understood ina probabilistic sense: it says that the number of states where (10) is not satisfiedshould have a slower growth as a function of N than the total number of states |T N

6What is relevant for our current purposes is always a configuration space path integral, andnot one in phase space.

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(which grows exponentially with N for a given spatial topology). Heuristically onecan view (10) as expressing that in the large-N limit the matrix elements T N |T |T ′N ′depend not on the two quadrangulations separately, but only on a suitably defined“distance” between T N and T ′N ′ , similar to the way the integral

dxf (x−y) over the

real line is independent of y. In principle it is a combinatorial problem to show that

(10) is valid, but we have not yet produced such a proof.An important consequence of property (10) is that the completeness relation (8)continues to hold for the area states |N ,

N 2|T t1+t2 |N 1 =N

N 2|T t2|N N |T t1|N 1, (11)

where we have again assumed that N 1, N 2 and N are all large. We therefore stillhave a transfer matrix formalism, but with the transfer matrix T acting only on thesubspace spanned by the linear combinations {|N } of the original Hilbert space.Obviously, the task of diagonalizing the transfer matrix on this reduced space is

considerably simplified compared with the original problem.From now on, we will focus our interest on solving the combinatorics of a singlesandwich geometry. We can rewrite relation (6) in an obvious notation as

N 14|T |N 41 = e−c0−(λ−k)(N 41+N 14)

N max22

N 22=N min22

N (N 41, N 14, N 22) e−( 12λ+k)N 22, (12)

where N (N 41, N 14, N 22) =

T N 41 ,T N 14

N (T N 41, T N 14, N 22) (13)

denotes the total number of quadrangulations of the space-time between t and t+1,including sums over the connectivities of the spatial boundary geometries at t andt+1. One may think of (13) as describing the combinatorics of quadrangulating asandwich geometry with free boundary conditions, except for the areas of the twoboundaries which are kept fixed.

In order to simplify the combinatorics further, we introduce boundary cosmolog-ical constants Λi and Λf associated with the initial and final boundaries at t andt+1. They do not have an immediate physical interpretation and should simply bethought of as convenient book-keeping devices that will be set to zero at the end7.They give rise to an additional term

∆S = Λia2sN 41 + Λf a2sN 14 ≡ ziN 41 + zf N 14 (14)

in the action (3), which will allow us to introduce an asymmetry in the couplingconstants multiplying the two areas N 41 and N 14. We can use the dimensionless

7This is analogous to the introduction of external sources into a quantum field-theoretic pathintegral in order to obtain Green’s functions through functional differentiation.

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boundary cosmological terms zi and zf to obtain the Laplace transform of the trans-fer matrix with respect to both the initial and final areas, N i ≡ N 41 and N f ≡ N 14,namely,

zf |T |zi =N i,N f

e−ziN i−zf N f N f |T |N i. (15)

The matrix elements N f |T |N i are calculated for zi = zf =0. From a combinatorial

point of view, N f |T |N i counts the three-geometries G for given numbers N i and

N f , each with relative weight e−(12λ+k)N 22, and zf |T |zi plays the role of a generating

function for these numbers. It is usually much easier in combinatorial problems tocalculate the generating function rather than the actual numbers. This is well illus-trated by both Lorentzian and Euclidean simplicial quantum gravity in dimensiontwo, where boundary cosmological constants are introduced in an analogous man-ner, and where the associated generating functional greatly simplifies the countingof geometries.

The reason for the simplification is the fact that by going to

zf

|T

|zi

we have

achieved totally free boundary conditions, since the constraints of fixed areas N iand N f have been lifted. On the other hand, no information has been lost, since

we can in principle always rederive N f |T |N i by an inverse Laplace transformation

from zf |T |zi. Given the one-step propagator N f |T |N i, we can finally obtain thepropagator G(N i, N f , t) for arbitrary times by iterating according to (11).

4 Extracting the Hamiltonian

Once the matrix elements N f |T |N i are known, one may try to extract the con-

tinuum Hamiltonian operator H of the system by expanding them in the latticespacing a and then taking a → 0,

N f |T |N i = N f |e−aH |N i = N f |

1 − aH + O(a2)|N i. (16)

In this way one obtains the quantum Hamiltonian in the “N -representation”. It isalso possible (and usually easier) to extract H from the Laplace transform of the one-step propagator, zf |T |zi (yielding the Hamiltonian in the “dual” z-representation).

Let us illustrate this by a concrete calculation in 2d Lorentzian gravity, where thespace-time has topology S 1 × [0, 1]. In this case, the generating functional zf |T |ziis known explicitly8 [6],

zf |T |zi = log

(1 − e−(λ2+zi))(1 − e−(λ2+zf ))

1 − e−(λ2+zi) − e−(λ2+zf )

. (17)

8In line with our construction in dimension three, we consider here the symmetric propagatorof [6] with unmarked boundary loops.

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The relevant bare couplings in two dimensions are the bulk cosmological constantλ2 and the boundary cosmological constants zi and zf . They are related to therenormalized continuum coupling constants Λ2, Λi, Λf by

λ2 = log 2 +1

2Λ2a2, zi,f = Λi,f a. (18)

It is straightforward to expand (17) to lowest non-trivial order in the lattice spacinga and thus obtain the matrix elements Λf |H |Λi. We have

Λf |T |Λi = analytic − log(Λi + Λf ) − aΛ2 − 1

2 (Λ2i + Λ2

f )

Λi + Λf + O(a2), (19)

where “analytic” refers to a constant term and to terms linear in Λi and Λf . Byan inverse Laplace transformation we can change variables from the boundary cos-mological constants Λi,f to the conjugate (continuum) length variables Li,f of theboundaries. To first order in a, one finds

Lf |T |Li =

i∞

−i∞

dΛi

2πi

i∞

−i∞

dΛf

2πieΛiLi+Λf Lf Λf |T |Λi (20)

=1

Liδ(Li − Lf ) − a

− d2

dL2f

+ Λ2

δ(Li − Lf ),

where we have ignored the analytic terms which lead to non-propagating terms of the form δ(Lf )δ(Li). The appearance of such non-universal terms is familiar fromthe transfer matrix of 2d Euclidean gravity and can be accounted for by a carefultreatment of the boundary conditions at L = 0 [4, 21, 22].

In complete analogy with the definition given in footnote 4, the symmetry factor

of a spatial S 1

-boundary of length L is given by C L = 1/L and there is an orthogonalbasis {|L} of the continuum Hilbert space, obeying the normalization conditions

L1|L2 =1

L1δ(L1 − L2),

∞0

dL |LLL| = 1. (21)

We deduce that the Hamiltonian operator in the “L-representation”, acting on func-tions ψ(L) = L|ψ, is

H = − d2

dL2L + Λ2L, (22)

which is hermitian with respect to the measure LdL, as it should be.In principle we would like to use the same strategy to determine the quantum

Hamiltonian H (A) and its spectrum also in the three-dimensional case. Of coursewe cannot be sure that this will lead to a simple differential operator as a functionof the spatial volume, as was the case in d = 2, (22). Even in two dimensions, thereare models inspired by Lorentzian gravity where this is not the case [26]. However,in the 2d cases studied so far it has always been possible to turn H into a localdifferential operator by transforming to a variable different from L.

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Figure 2: A piece of a typical quadrangulation at t + 1/2. The three types of squares made from solid and dashed lines arise as sections of the (4,1)-, (1,4)- and(2,2)-building blocks.

5 The matrix model correspondence

We will now relate the Laplace-transformed one-step propagator (15) to a matrixmodel. This will be done by showing that the two-dimensional configurations asso-ciated with a slicing at half -integer t of a sandwich geometry appear as terms in theperturbative expansion of a hermitian two-matrix model with ABAB-interaction.(A similar observation for the original Lorentzian 3d simplicial model was alreadymade in [18].)

Imagine a one-step geometry G obtained by gluing the three types of buildingblocks of Fig. 1. The intersection of this three-geometry with the spherical constant-time hypersurface at t + 1

2

can be visualized as a pattern of squares, whose edgescorrespond to the intersections of the time-like triangular faces of the 3d buildingblocks with this surface. Let us distinguish the two cases where the time-like trianglehas its base either in the t- or the t + 1-quadrangulation by drawing its intersectionat t+1

2as either a solid or a dashed line. Thus a (4,1)-pyramid gives rise to a square

of solid edges, a (1,4)-pyramid to one of dashed edges and a (2,2)-tetrahedron toa square with alternating solid and dashed edges. The way in which these two-dimensional building blocks appear in the S 2-quadrangulation at t+ 1

2is that they

can be glued to each other only along pairs of edges of the same type.The quadrangulation at t+ 1

2can thus be viewed as a double-line graph of the

kind illustrated in Fig. 2. This type of graph is generated in the large-M limit by

the two-matrix model

Z (α1, α2, β ) =

dAdB e−M tr ( 1

2A2+ 1

2B2−

α14

A4−α24

B4−β2ABAB), (23)

where A and B are Hermitian M × M matrices. By expanding the non-Gaussianpart of the exponential in powers of α1, α2 and β and performing the Gaussian

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i

j k

l

j k

li

B ji lk Blk A

B

A ji

i

j k

l

A ji

Bil

lk A

i

j k

l

j k

li

B ji lk Blk AA ji

i

j k

l

A ji

Bil

lk A

kjA

ilA

kjB

ilB

kj

Figure 3: Matrix-model representation of the building blocks at t + 1/2. The gluingrules for the squares are determined by the Gaussian integrations, AijAkl = δilδ jk ,BijBkl = δilδ jk , and AijBkl = 0.

integral we are led by Wick’s theorem to a successive gluing of the three kinds of squares described above if we make the index assignments as shown in Fig. 3.

As usual, the logarithm of the partition function of the model, M 2F (α1, α2, β ) =

log Z (α1, α2, β ), generates only connected quadrangulations, and taking the large-M limit will select those with S 2-topology. In principle F may be expanded in a powerseries in M −2, with higher-order contributions corresponding to quadrangulationsof higher genera. Although we are mainly interested in the spherical limit, anythingwe say could be repeated for higher-genus surfaces, thus relating to 3d gravity onspace-times with topology Σ(g)×[0, 1]. We can now write out the generating functionF as an explicit power series,

F (α1, α2, β ) =

N 41,N 14,N 22

˜ N (N 41, N 14, N 22) αN 411 αN 14

2 β N 22, (24)

where ˜ N (N 41, N 14, N 22) denotes the number of (connected) spherical quadrangula-tions described above, including symmetry factors.

Comparing the form of (24) with the previous expressions (12) and (15), andmaking the identifications9

α1 = ek−λ−zi , α2 = ek−λ−zf , β = e−(12λ+k), (25)

one could be tempted to conclude that

F (α1, α2, β )?

= zf |T |zi. (26)

However, this is not correct, since the number of configurations generated by thematrix model is strictly larger than those obtained from the Lorentzian gravitymodel, that is,

˜ N (N 41, N 14, N 22) > N (N 41, N 14, N 22). (27)9 Adopting (25), the limit of vanishing boundary cosmological constants zi,f corresponds to

setting α1=α2. Unfortunately we cannot put α1 =α2 before having extracted the matrix elementsN f |T |N i from F (α1, α2, β ).

13



Figure 4: Vertices of φ4-graphs dual to spatial quadrangulations.

This difference is intimately connected to the fact that in the gravity case, theallowed configurations are not so much two-dimensional structures per se as sectionsof larger, three-dimensional objects whose three-dimensional manifold structure is

encoded in the colouring (the dashed and solid lines) of the two-dimensional graph.The generalization inherent in the matrix model is best described in terms of thegraphs dual (in a two-dimensional sense) to the quadrangulations. In this more con-ventional picture, the terms tr A4, tr B4 and tr ABAB are represented by four-valentvertices (placed at the centres of the squares of the original 2d quadrangulation) withfour outgoing solid lines, four dashed lines or alternating solid-dashed lines (see Fig.4). The labels A and B are now associated with the dual solid and dashed edgesconnecting pairs of such vertices.

In order to discuss the regularity conditions that must be satisfied by such adual graph to qualify as (the dual of) a section of a Lorentzian three-geometry, wedefine a A-loop (a B-loop) as a closed sequence of solid (dashed) dual edges with no

further solid (dashed) dual links in its interior. This interior region (which has theform of a two-dimensional disc whose boundary is the loop) we call a A-domain (aB-domain). A dual graph coming from a 3d Lorentzian geometry then satisfies thefollowing constraints (c.f. App. 2 of [18]):

(1) The two subgraphs formed from only A-edges and only B-edges must each beconnected.

(2) The two separate A- and B-subgraphs can have neither tadpoles nor self-energy subdiagrams. This ensures that they are associated with regular 2dsimplicial manifolds at the times t and t + 1 respectively.

(3) The intersection of any pair of A- and B-domains cannot be multiply con-nected. This implies that any pair of vertices, one at time t and one at timet + 1, of the original quadrangulation cannot be connected by more than onetime-like link.

(4) The (one-dimensional) intersection of a A-loop with a B-domain (and vice

14



versa) must be either empty or simply connected. This implies that any threevertices of the original quadrangulation cannot belong to more than one time-like triangle.

Thus we see that the dual graphs coming from the matrix model are considerablymore general than those associated with 3d simplicial space-times. The generaliza-tions occur in several ways. The matrix-model graphs can have arbitrary numbersof disconnected spherical A- and B-subgraphs (the only requirement being that thecombined graph is spherical and connected) and each of these may contain (gen-eralized) self-energy diagrams and tadpoles. Furthermore, the A- and B-loops canfreely meander around each other, with arbitrary numbers of mutual intersections.

We conjecture that the conditions (2)-(4) are not important in the sense that theirimplementation or otherwise will not affect the continuum properties of the model.We think that they constitute merely a 3d generalization of the universal behaviouralready observed for 2d matrix models. In that case the inclusion of tadpole andself-energy subgraphs leaves the continuum limit unchanged (although it implies an

enlargement of the configuration space from genuine 2d simplicial manifolds to moregeneral 2d combinatorial complexes).

We have checked numerically that the same happens in our 3d model. Moreprecisely, we have performed simulations for the original simplicial model with tetra-hedral building blocks of [15, 18, 16] where one can formulate conditions completelyanalogous to (1)-(4) above. Dropping then the constraints (2)-(4), but keeping (1),we found that the key results of [18] remained unaffected.10

The status of condition (1) is different. Recall that in the original representationone obtains the geometry at time t (at time t + 1) from the quadrangulation att + 1/2 by shrinking all dashed (solid) links to zero. The A- and B-graphs of thedual picture at t + 1/2 are of course individually precisely the duals of these two-

geometries at integer times. What does it imply for the spatial quadrangulation att if the dual A-graph is disconnected? Using the same “shrinking-prescription”, itcannot separate into several pieces, but it does degenerate in the sense of forming anumber of connected spherical graphs which touch each other pairwise only in singlepoints. The resulting two-dimensional space is therefore no longer a manifold, buta branched tree of such spherical components.

This phenomenon is illustrated in Fig. 5. The pictures at the top show two quad-rangulations of a two-sphere at time t+1/2 made of the three types of square-shapedbuilding blocks. As two-dimensional configurations they look rather innocent andregular. However, they are pathological from a three-dimensional point of view.

10However, one interesting change did occur: in [18] we observed a first-order phase transition forlarge k (there called k0) to a phase where successive spatial slices decouple. We viewed the presenceof this phase as a discretization artifact irrelevant for the continuum physics. This interpretationis corroborated by our recent simulations without the constraints (2)-(4), where the large-k phaseis simply absent. It implies that the generalized model gives in some sense a better representationof continuum physics. This is reminiscent of 2d simplicial quantum gravity, where the continuumlimit is approached much faster if self-energy and tadpole graphs are allowed.

15



(a) (b)

Figure 5: Examples of matrix-model configurations at t + 1/2 which are not allowedin the original Lorentzian gravity model and which result in geometries with worm-

holes at time t. Shrinking the dashed links to zero, one obtains the two-geometriesat the bottom. The thick dashed lines in the quadrangulations at the top are con-tracted to touching points or to points along one-dimensional wormholes.

Drawing the dual graphs, one finds in both cases that the A-subgraph, made of solid dual edges, consists of two components. Each component is dual to a quad-rangulation of a two-sphere by four (solid-line) squares. When shrinking away thedashed links of the original quadrangulation, the “necks” (indicated by the thickdashed circular lines) between the two spheres are gradually pinched to points. If there is just one neck, the two spheres will touch in a point (Fig. 5a). If there

are several concentric necks, with annuli of dashed-solid squares (corresponding toclosed rings of (2,2)-tetrahedra) in between, this process generates one-dimensional“wormholes”, as illustrated in Fig. 5b.

Similar so-called “touching”-interactions have been studied in the context of two-dimensional matrix models [27, 28], and are closely related to these wormholes (seeSec. 6 for a detailed discussion). Although configurations of this type are already

16



present in ordinary matrix models of Euclidean 2d gravity, the explicit introductionof such an interaction term in the action (and an associated coupling constant) al-lows us to increase their weight and thereby control their abundance. Dependingon the value of the coupling constant, one can obtain either the ordinary universalbehaviour of two-dimensional Euclidean gravity, or a modified critical behaviour,

described in more detail below. We will see that in our 3d-gravitational reinterpre-tation of the matrix model the presence of wormholes and touching-interactions isgoverned by the value of the bare gravitational coupling constant k.

Let us finally note that the above discussion about degenerate geometries couldbe repeated verbatim for the cubic matrix model with partition function

Z (α1, α2, β ) =

dAdB e−M tr ( 1

2A2+ 1

2B2−

α13

A3−α23

B3−β2ABAB), (28)

which is associated with the original 3d simplicial gravity model described in [15, 16](with (3,1)- and (1,3)-tetrahedra instead of (4,1)- and (1,4)-pyramids). The only

reason why we prefer to use the quartic model defined by (23) is that it has beensolved for α1 = α2 [20]. We would expect from universality arguments that themodels given by (23) and (28) lead to the same continuum physics.

6 Reinterpreting the matrix model

We have seen in the previous section that 3d Lorentzian “sandwich” geometriescan be put into correspondence with a subclass of 2d graphs generated by a quarticmatrix model with ABAB-interaction. We have also argued that this generalizationis potentially relevant, in the sense of the two models having a different phase

structure.The phase structure of the three-dimensional Lorentzian model has been inves-tigated numerically in [18, 19]. For given gravitational coupling k, there is a criticalvalue λc(k) which is to be approached from above, that is, from the region λ > λc

of the cosmological constant where the partition function converges. In the orig-inal simulations we also observed a first-order transition for large k when movingalong the critical line λc(k). As already mentioned in footnote 10, this transitionis a discretization artifact which disappears when one relaxes some of the mani-fold constraints. What emerges as the phase structure for 3d Lorentzian gravity istherefore simply that of a single phase (at least in the range of coupling constantsaccessible to our computer simulations) where the taking of the continuum limit co-

incides with the tuning of the cosmological constant to its critical value, much as intwo-dimensional models of quantum gravity. The role of the gravitational couplingk is merely to set an overall scale for the system, without affecting its continuumproperties, as we have argued in [18].

What we want to explore presently is the physical interpretation in terms of three-geometry of the generalized model (where multiply-connected A- and B-graphs

17



α

0.05 0.1 0.15 0.2 0.25

0.05

0.1

β

κ > 1κ = 1

κ < 1

Figure 6: The phase diagram of the ABAB-model, according to [20], with the critical

line and the critical point at α = β =1

4π (corresponding to κ = 1). One way of approaching the critical line is through fine-tuning of the cosmological constant λalong lines of constant k, shown as dashed curves. To end up at the critical point,one should move along the line of constant kc = −1

3ln β c ≃ 0.85.

are allowed to occur as the spatial sections at half-integer times) and of its phasestructure. This is of course the two-matrix model defined by eq. (23). Let us considerfirst the case of vanishing boundary cosmological constants, zi,f = 0. Although wewill not be able to construct the Hamiltonian or the propagator (c.f. footnote 9),we can nevertheless discuss the phase diagram of the model. We will comment later

on the general matrix model with zi, zf = 0.Physically this choice of couplings implies that we are studying the geometricfluctuations between two successive spatial slices with free boundary conditions.As was clear from the computer simulations in [18], the behaviour of this one-stepsystem determines the phase structure of the discretized theory. Other than that,the choice zi,f =0 has the great advantage that the corresponding matrix model hasalready been solved. Details of the solution can be found in the paper of V. Kazakovand P. Zinn-Justin [20]. Their analysis of the matrix model with α1 = α2 = α canbe summarized as follows. Let us fix a ratio κ = α/β and gradually increase thevalues of α and β away from zero. The large-M limit of the matrix model is definedfor sufficiently small α and β , and the model becomes critical at a point (β c(κ),

αc(κ) = κβ c(κ)), giving rise to a critical line in the (β, α)-plane (see Fig. 6). Alongit one finds two phases, separated by a second-order phase transition at κ = 1,with αc(1)= β c(1). In the context of two-dimensional gravity, they were given thefollowing interpretation [20]: the phase of κ > 1 with αc(κ) > β c(κ) can be viewed asordinary two-dimensional Euclidean gravity, with central charge c =0. In this phasethe relation between the length l of a “typical” loop of A-links (or B-links) and the

18



area A of the two-dimensional surface enclosed by the loop is l2 ∼ A. By contrast,in the other phase (which also has c = 0), the scaling is anomalous, l4/3 ∼ A.

One can understand these two situations by simply looking at the two limitsβ = 0 and α = 0. By setting β = 0 we switch off the ABAB-interaction, leadingto a decoupled system of two φ4 one-matrix models, whose critical behaviour is

individually that of an ordinary 2d Euclidean gravity system. Setting α = 0, weobtain a matrix model whose only interaction comes from the term tr ABAB. Thismodel was first solved in [29] by mapping it to the so-called dense loop phase of theO(1)-matrix model on random graphs.11

6.1 Intermezzo: touching-interactions

We will now reinterpret the results described above in the context of three-dimensio-nal Lorentzian gravity. Recall our earlier discussion in Sec. 5, where we showed thatfrom a three-dimensional point of view a generic feature of the matrix-model config-urations is the presence of touching points and wormholes in the associated spatial

slices. The matrix model therefore describes transitions between two-geometrieswith a tree-like structure, consisting of regular quadrangulations of two-spheres,pairwise connected by one-dimensional wormholes of length l ≥ 0 (Fig. 5).

Let us for a moment revert back to a pure matrix-model language. For the sakeof definiteness, we will concentrate on a quadrangulation at time t made from solidsquares. We will now show that the effective weight associated with the mutualtouching points of the two-spheres making up the tree-configuration is obtained byintegrating out the B-matrices. While this cannot be done explicitly as long as thetr B4 term is present, it is instructive to do the integration for the general matrixmodel (23) with α2 = 0.12 In this case, the action is Gaussian in the B-matrix and

we obtain

Z (α1, α2 = 0, β ) =

dA dB e

M

− 12

tr (A2+B2)+α14

tr A4+ 12β tr ABAB

=

dA

det(I − βAT ⊗ A)

− 12

eM

− 12

tr A2+α14

tr A4

=

dA e

M

− 12

tr A2+α14

tr A4+ 12M

∞k=1

βk

ktr Aktr Ak

, (29)

which now describes a particular one-matrix model. Terms like tr Aktr Ak are usu-

ally referred to as “touching-interactions”. How can their effect be visualized in11 The O(1)-model on random graphs is known to correspond to c = 0 quantum gravity, but

with two distinct phases [30, 31, 32], not unlike those of the ABAB-model. In a continuuminterpretation, these result from assigning different boundary operators to the theory. The diluteloop phase corresponds to ordinary 2d Euclidean quantum gravity where boundary lengths scaleaccording to l2 ∼ A, while in the dense phase one has l4/3 ∼ A [31, 32].

12For α2 = 0 there will be other touching interactions beyond the ones described below.

19



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 01 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 10 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1

0 0 0 01 1 1 1 0 0 0 01 1 1 100000001111111 0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 01 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 100000001111111

0 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1

0 0 0 01 1 1 1 0 0 0 01 1 1 10000000111111100000001111111

0 0 0 01 1 1 1 0 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 10 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1 0 0 0 01 1 1 10 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1

Figure 7: A typical configuration generated by the matrix model (23) for α2 = 0. By

integrating out the B-matrices one obtains touching terms tr Ak

tr Ak

(here, k = 4and 6). Their effect is to identify the boundary links of the solid-square componentspairwise across the “gaps” formed by the solid-dashed squares, as described in thetext.

geometric terms? Firstly, in the same way as tr A4 represents a square of solidedges, a term tr Ak in the action can be thought of as a k-gon. Expanding theaction and performing Wick-contractions corresponds to gluing such polygons to-gether by identifying their links pairwise. Imagine that during this process we hadconstructed a surface glued from various polygons, with a boundary consisting of klinks. By performing the appropriate Wick-contractions, we could now glue a k-gon(represented by tr Ak) to this boundary and close off the surface.

By extension, we can determine the effect of a term like tr Aktr Ak. Imaginea pair of surfaces made from polygons, both with a boundary of length k. Bycontracting with tr Aktr Ak, we can close off the two surfaces. Because this happenssimultaneously for both of the surfaces, we associate by this process a k-gon of thefirst surface with another k-gon of the second surface, which we may think of as atouching point between the two geometries (for example, located at the centres of the k-gons).

This is illustrated by Fig. 7, which shows a typical quadrangulation at t + 1/2appearing in the matrix model with α2 = 0 (so no dashed squares are present),

before performing the B-integration. The shaded areas represent components madeof solid squares, where the outer boundary should be thought of as a single point,since the entire geometry is spherical. Integrating out the B-matrices has preciselythe effect of generating terms tr Aktr Ak and identifying the solid boundary linkspairwise (each link with its partner lying opposite in a solid-dashed-solid-dashedsquare). Within the one-matrix model it is of course a matter of convention how

20



one wants to think of the pairwise associations introduced by the terms tr Aktr Ak,whether as microscopic touching points between the two closed-off surfaces or simplyas regular gluings of one boundary of length k to another. However, it is naturalto think of them as creating a connection between the surfaces. There is a factor1/M associated with each touching interaction, and it is well-known that to leading

order in M one obtains through “touching and connecting” only two-geometrieswith an overall spherical topology, and no spaces of higher genus. Collapsing all thegenuinely two-dimensional parts of this two-geometry to one-dimensional lines, wewould obtain a branched polymer, i.e. a one-dimensional tree structure without anyclosed loops.

What is our motivation for this excursion into touching-interactions? The generalABAB-model does not in itself suggest an immediate physical interpretation interms of spatial or space-time geometry. However, with the benefit of hindsight,one could have taken the presence of the limiting case β = 0 (where one has twoindependent copies of Euclidean 2d gravity) as an indication that switching on the β -coupling will lead to a theory of two interacting two-geometries. The situation where

such an interaction occurs naturally is of course when the two-geometries appear asneighbouring embedded spatial slices within a 3d space-time, with their interactiondictated by the 3d Einstein action. Given the three-dimensional interpretation, it isthen natural to determine the effect of the neighbouring two-geometry by integratingout the corresponding degrees of freedom in the partition function (for instance, theB-matrices). As we have seen, this effectively introduces touching-interactions in a2d slice which are dictated by β , where β in turn depends on the coupling constantsλ and k of three-dimensional quantum gravity.

Another related point that follows from the three-dimensional interpretation isthat it gives us a precise geometric way of obtaining a two-geometry at time t (or

t + 1) from a general matrix-model configuration at time t + 1/2 by the shrinking-prescription already mentioned in Sec. 5. This may be visualized as a continuousprocess where the lengths of either the dashed or the solid links are gradually shrunkto zero. Although this prescription was originally invented to describe the well-behaved geometries of 3d Lorentzian quantum gravity, it works just as well for themore general configurations generated by the matrix model. We have already seenexplicit examples in Sec. 2, where this led to the creation of spatial wormholes (Fig.5).

What is important to notice is that from a three-dimensional point of view,such wormholes and the touching-interactions described in this subsection are reallytwo sides of the same coin. That is, taking a general configuration of the ABAB-

model which is not allowed in the original Lorentzian model, one will find both typesof degeneracy when applying the shrinking-prescription. We illustrate this with asimple matrix-model graph in Fig. 8. We start at time t + 1/2 from an extendedversion of Fig. 5b, where the shaded areas indicate regular dashed quadrangulations.Since these will all vanish when we shrink away the dashed edges, we obtain thesame wormhole geometry at time t as before. On the other hand, if we now shrink

21



In phase II, the touching- and the tr A4-interactions coexist and compete. Atypical two-geometry consists of many spherical 2d “baby”-universes of all sizes,which are connected to each other at touching points, thus forming a “blown-up”branched polymer14. As was first realized in [34], the critical behaviour resultsfrom an interplay between the properties Euclidean gravity and those of branched

polymers (see also [35]).Quite surprisingly, all critical aspects of this phase can be understood fromcontinuum quantum Liouville theory (which is usually thought to describe onlyphase I) [36, 37, 38]. Let us review this briefly. Consider a conformal field theoryin dimension two with a fixed background metric g, and let Φ be a spinless primaryfield with scaling dimension ∆

(0)Φ . Then the “one-point” function for Φ scales as

F (0)

Φ (A) =

d2ξ

g(ξ) Φ(ξ)

CF T

∼ A1−∆(0)Φ , (30)

where

A= Σ d2ξ g(ξ) is the area of the underlying manifold Σ, and the ex-

pectation value is taken in the conformal field theory. Coupling this theory to 2dEuclidean quantum gravity, the metric g will be allowed to fluctuate. We can decom-pose a general metric g into g = eφg, where φ(ξ) is the conformal field. In conformalgauge, the integration over the fluctuating metric becomes an integration over theconformal factor φ, weighted by the Liouville action S L[φ], while the spatial integralappearing in the one-point function is changed according to

d2ξ

g(ξ) Φ(ξ) −→

d2ξ

g(ξ) eβ Φφ(ξ)Φ(ξ). (31)

The dressing exponent β Φ is determined by requiring that the “dressed” operator

eβ Φφ

Φ have conformal dimension dimension (1,1), so that it can be integrated overthe two-dimensional surface without breaking conformal invariance. This leads tothe relation

β Φ(β Φ + Q) = −2 + 2∆(0)Φ , Q =

25 − c

3. (32)

The special case of the unit operator Φ = 1 corresponds to ∆(0)1

= 0, with dressingexponent β 1. The one-point function for the gravity-coupled theory is defined by

F Φ(A) =

d2ξ

g(ξ) eβ Φ φ(ξ)Φ(ξ)

QG

:= Dφ e

c−25

48π2 S L[φ] δ(

d2ξ√g eβ 1φ−A)

d2ξ √g eβ Φ φ ΦCF T

Z (A), (33)

where Z (A) is the partition function with fixed area A (enforced by including thesame δ-function that appears in the numerator of (33)). By requiring that F Φ(A) be

14Further information about the properties of branched polymers can be found in [ 33].

23



independent of the fiducial background metric g, a change of integration variablesφ → φ + 1

β 1log A leads to

F Φ(A) = Aβ Φ/β 1 F Φ(1). (34)

In analogy with (30), we now define the critical exponent ∆Φ of Φ for the gravity-coupled theory byF Φ(A) ∼ A1−∆Φ. (35)

Combining this with relation (34), one derives

∆Φ = 1 − β Φβ 1

. (36)

For our present purposes it is important to note that eq. (32) has two solutions,namely,

β (±)Φ = −(25

−c)

± (25−

c)(1−

c + 24∆(0)Φ )

6Q . (37)

In conventional quantum Liouville theory one works with β (+)Φ because this choice

ensures that in the “classical” limit c = −∞ (classical in the sense that the fluctua-tions of the Liouville field in (33) are completely surpressed) the scaling dimensionsof the model without gravity coupling are recovered,

∆Φ → ∆(0)Φ for c → −∞, (38)

as can easily be verified from formulas (37) and (36).It is a remarkable fact that all critical exponents which can be calculated in

phase II of the matrix model with touching-interactions are obtained by replacingβ

(+)Φ → β

(−)Φ , changing the scaling relations according to

F (+)

Φ (A) ∼ Aβ (+)Φ /β

(+)1 −→ F

(−)Φ (A) ∼ Aβ

(−)Φ /β

(+)1 . (39)

The implications of this prescription can be illustrated by considering the simplestboundary operator,

ℓ =

∂ Σ

ds

e(s), (40)

which measures the length of the boundary of the two-dimensional manifold Σ as a

function of the induced metric e(s) on ∂ Σ. Unlike in matrix models, it is somewhatawkward to introduce boundary operators like (40) in Liouville theory. Nevertheless,ℓ can be treated along similar lines as the operator Φ in (31) above. The expressionone obtains for the gravitationally dressed version of the length operator is

ℓ =

ds

e(s) eβ (+)ℓ

φ(s), β (+)ℓ =

1

2β

(+)1

. (41)

24



From the point of view of conformal field theory it is rather surprising that the“naive” value β ℓ = 1

2 β 1 (i.e. the boundary scales like the square root of the bulk)is correct. However, as can be read off from (34), this does lead to the canonicalscaling

ℓ(+)(

A)

QG

∼ Aβ (+)ℓ

/β 1 =

A1/2 (42)

already known from matrix-model considerations.To determine the scaling relevant in phase II, we should now replace the operator

β (+)ℓ . It turns out that the correct substitution for this particular boundary operator

is given by

β (+)ℓ −→ β

(−)ℓ =

1

2β

(−)1

. (43)

This leads to the scaling behaviour

ℓ(−)(A)QG ∼ Aβ (−)ℓ

/β (+)1 = A3/4, (44)

implying that the boundary scales anomalously in terms of the area. This anomalousscaling is clearly a reflection of the additional fractal structure introduced by thetouching-interactions. Note that it coincides with the spectral dimension of branchedpolymers [39, 40].

The scaling (44) coincides with the scaling behaviour observed in the ABAB-matrix model in the phase with κ < 1, and mentioned already earlier in Sec. 6. Thiscorroborates our interpretation of this model in terms of interacting two-geometries.Furthermore, it agrees with the relation between loop length and area found in thedense phase of the O(1)-model, where the replacement (43) was first noticed [32].

Lastly, the critical behaviour in phase III of the matrix model with touching-interactions is characterized by a complete dominance of branched polymers. The

size of the individual spherical components making up the 2d universe never exceedsthe cutoff scale. This phase does not seem to have an analogue in the ABAB-matrixmodel.

6.3 Summary of Sec. 6

Let us summarize what we have learned about the three-dimensional geometricinterpretation of the ABAB-model. To this end, we will first translate its phasediagram (for α1 = α2) to the k-λ-plane of the gravitational couplings. We deducefrom (25) that

α

β 2= e3k and

1

GN ∼k =

1

3log

κ

β c(κ), (45)

where the second relation holds along the critical line (the points where a non-trivial continuum limit exists). The shaded areas in the two phase diagrams of Fig.9 indicate the region in which the partition function converges. The dashed curvesin both diagrams show an approach to the critical line along a curve of constant k.Moving from left to right along the critical curve of the left phase diagram (so that

25



α

0.15 0.25

β

0.1

0.05

0.20.05 0.1

~

~

3

1

4

k

λ

0

Figure 9: Translating the phase diagram of the ABAB-matrix model in terms of the original coupling constants α and β (left) to the couplings k and λ associatedwith its interpretation in terms of 3d Lorentzian gravity (right).

β increases and κ decreases) translates into a motion from right to left along the

critical line of the “gravitational” phase diagram (that is, towards a smaller k or alarger bare Newton’s constant GN ).

In terms of space-time geometry, at the point GN = 0 (corresponding to β = 0)subsequent spatial slices are completely decoupled. No information can be propa-gated in time and the system has no interesting three-dimensional properties. In-creasing GN away from zero, we enter the phase κ > 1 of the matrix model. Interac-tions between neighbouring spatial slices of the 3d universe become possible15, andthe geometrical properties of a typical spatial slice at integer-t resemble those gen-erated in 2d Euclidean quantum gravity. It is very suggestive to identify this phasewith that of Lorentzian gravity defined in [15, 16], since the simulations of [18, 19]revealed both the existence of extended three-dimensional space-times (indicative

of correlations in time-direction), and evidence that the Hausdorff dimension of thespatial slices is dH = 4, in agreement with that of 2d Euclidean gravity.

Apparently, the phase with κ ≤ 1 cannot be realized in the Lorentzian manifoldmodel because it does not allow for the creation of “wormholes”16. By contrast,no such restriction exists in the ABAB-matrix model, and by gradually increasingNewton’s constant GN such wormhole configurations become energetically morefavoured. Looking at spatial slices in the phase κ ≤ 1, a typical two-geometry willconsist of many smaller universes connected to each other by cutoff-size wormholes.

15We cannot be more specific about the nature of the interactions before having analyzed thetransfer matrix of the model in more detail.

16At least, we have so far not seen any evidence of this phase or the associated second-ordertransition in the computer simulations.

26



7 The general ABAB-matrix model

While the phase structure of the matrix model of (generalized) 3d Lorentzian gravitycan be understood from the KZ-solution for the special case α1 = α2 of the ABAB-matrix model, the construction of the quantum Hamiltonian requires that we perturb

away from α1 = α2. It is not difficult to generalize the ansatz of [20] to this situation,using the techniques developed in [41]. One can write down a set of singular integralequations which generalize those solved in [20]. Furthermore, using results fromthe general theory of singular integral equations one can prove that the solution isunique.

Although we have at this stage nothing much to say about the explicit solutionfor general α1 and α2, it may be worth pointing out a connection with yet another2d statistical model. As mentioned in [20] and studied in more detail in [42], theABAB-matrix model with α1 = α2 can be mapped to an eight-vertex model definedon a random four-valent lattice (in the dual picture). The map is simply given byX = A + iB. Using the same transformation, one can map the general ABAB-model

to a sixteen -vertex model with partition function

Z (α1, α2, β ) =

dX dX † e

−N tr

12X†X−bX2X†2− c

2(XX†)2−d

4(X4+X†4)−f (X3X†+XX †3)

,

where the constants b, c, d, and f are given by

b =α1 + α2 + 2β

16, c = d =

α1 + α2 − 2β

16, f =

α1 − α2

16. (46)

Since the eight-vertex model on a regular square lattice has already been solved, itmay not be very surprising that the restricted ABAB-model can be solved too. Thiswould be in line with the general observation that it is often simpler to solve mattermodels on random dynamical lattices (associated with 2d quantum gravity) ratherthan on regular ones. By the same token, one should maybe not be too discouragedby the fact that the general sixteen-vertex model has not been solved on a regularlattice. This is also not what is needed here, because of the particular form of theparameters (46).

8 Taking the continuum limit

We have already described in some detail in Subsection 6.3 how the phase structureof the matrix model with ABAB-interaction can be reinterpreted from the point of view of three-dimensional geometry, and how this fits into our previous investigationsof three-dimensional Lorentzian quantum gravity. One further point we would liketo make concerns the nature of the continuum limit in the gravitational model,and what the newly established relation with the matrix model can add to ourunderstanding of it.

27



In dynamically triangulated models of quantum gravity one usually performsthe continuum limit by fixing the bare (inverse) gravitational coupling constant kand fine-tuning the bare cosmological constant λ to its critical value λc(k). Sincethe space-time volume is conjugate to the cosmological constant in the action, sucha fine-tuning corresponds to taking the lattice volume N tot = N 14 + N 41 + 1

2 N 22 to

infinity, and can be viewed as an additive renormalization

λ = λc(k) + ∆λ, ∆λ = Λ3aν , (47)

of the cosmological constant, where ∆λ should be related to the continuum cosmo-logical coupling constant Λ3 by a suitable scaling.

The analogue of this procedure in a lattice field theory, for example, the Isingmodel defined on a hypercubic N d-lattice, would be to take the infinite-lattice limit.However, this may not lead to a continuum limit. In the case of the Ising model,in order to obtain such a limit, the coupling constant β (the inverse temperature)must always be fine-tuned to a critical value, even if the lattice volume was infinite.By analogy, one might therefore expect that also in simplicial gravity models, thegravitational coupling k had to be fine-tuned to a critical value kc to arrive at aninteresting continuum theory.

However, this is not the only possible scenario. Consider another hypercubiclattice model in d dimensions, one with a lattice scalar field φn and the simpleGaussian action

S [φ] =n

di=1

(φn+ei − φn)2. (48)

For this model, one automatically obtains the continuum Gaussian field theory bytaking the limit of infinite lattice volume and rescaling the lattice spacing to zero.

The same situation is encountered in two-dimensional simplical quantum gravity,both for Lorentzian and Euclidean signature, when formulated as a sum over tri-angulations with geodesic edge lengths determined by a single lattice spacing a.Fine-tuning the cosmological coupling constant λ(2) to its critical value accordingto

λ(2) = λ(2)c + ∆λ(2), ∆λ(2) = Λ2a2, (49)

or, equivalently, taking the infinite-volume limit and scaling a to zero, leads auto-matically to a continuum theory of 2d quantum gravity. In (49), Λ2 denotes thecontinuum two-dimensional cosmological constant, which has already appeared inour earlier discussion of Lorentzian 2d gravity (see eq. (18)).

It is easy to see that the taking of the continuum limit in the two-matrix modelwith ABAB-interaction falls into this latter category. According to [20], the con-tinuum limit is obtained by fixing the ratio κ= α/β (corresponding to straight linesthrough the origin in Fig. 6), and increasing α and β until the critical point isreached. This is a natural procedure to adopt, since the ratio α/β appears in theequations which determine the solution of the model. However, one could equallywell approach the critical curve αc(κ) = κβ c(κ) along any family of curves in the

28



2d conformal field theories with two-dimensional (boundary) surfaces in theories of 3d gravity. Examples are given by the recent entropy calculations in anti-de Sitter[43] and de Sitter space [44]. We cannot directly compare with any of these results,because our set-up is quite different. We are working in a geometric and not agauge-theoretic Chern-Simons formulation, our boundary surfaces are compact and

space-like, and our (bare) cosmological coupling constant is by necessity positive.Nevertheless, the ABAB-matrix model is suggestive of the presence of additionalsymmetries at the phase transition point.

As for the topology of our model, we have restricted ourselves to discussing space-times S 2 × [0, 1], since the solution of the ABAB-matrix model as presented in [20]is valid only for spherical topology. In order to make a comparison with other ap-proaches to (2+1) quantum gravity, it would be desirable to consider also surfaces of higher genus g, where the physical configuration space is described by a finite num-ber of Teichmuller parameters. Results obtained so far for the matrix model suggestthat its phase structure does not change for higher genus; the values of the criticalcoupling constants are independent of g, and most critical exponents are unaltered

since they refer only to the short-distance behaviour of the model. Nevertheless itwill be difficult to compare with a canonical reduced phase space quantization, say,because in the matrix model all modular parameters appear integrated over for agiven genus. It is possible one could find a way to probe the individual Teichm ullerparameters, although it seems already quite complicated to perform the necessarylarge-M expansion of the ABAB-matrix model using the character expansion of [20].

We have argued in this article that under certain assumptions the two-matrixmodel with ABAB-interaction describes three-dimensional Lorentzian quantum gra-vity. In fact, not only does it describe the “regular” Lorentzian quantum gravityformulated in [15, 18], but a more general theory where space can split into manycomponents connected by “wormholes”, resulting in a tree structure of spatial uni-verses forming a branched polymer. Starting at vanishing bare Newton’s constantGN , by increasing GN gradually we first find a phase of “weak gravity” which seemsto coincide with the phase of regular Lorentzian gravity seen previously in computersimulations. Eventually, we meet a second-order transition point beyond which liesa phase of “strong gravity” and large GN . From our original point of view of usingcausality as an effective regulator of quantum geometry, it is of course unclear towhat extent the space-times with an abundance of spatial wormholes found in thestrong-gravity phase are still acceptable in the path integral. In order to decide this

question, we need to get a better understanding of the genuinely three-dimensionalproperties of this system of quantum geometry.

30



Acknowledgements

All authors acknowledge support by the EU network on “Discrete Random Geom-etry”, grant HPRN-CT-1999-00161, and by ESF network no.82 on “Geometry andDisorder”. In addition, J.A. and J.J. were supported by “MaPhySto”, the Center of Mathematical Physics and Stochastics, financed by the National Danish Research

Foundation, and J.J. by KBN grant 2P03B 019 17.

Appendix

In this appendix we will calculate the Regge version of the Euclidean gravitationalaction (2), which we use in Sec. 2. It should be remembered that our geometriesare Wick-rotated versions of discrete Lorentzian (2+1)-dimensional space-times [16].For a change, we will do the angle calculations for Euclidean signature, and rotateback afterwards. Let the ratio of the squared lengths of time- and space-like linksbe given by a constant17 β > 1

2, such that l2

t = βl2s . The dihedral angles of (4,1)-

and (1,4)-pyramids around space- and time-like links will be denoted by θs and θt

respectively. Calling the corresponding dihedral angles of the (2,2)-tetrahedra φs

and φt, one derives the relations

cos2 θs = − cos θt =1

4β − 1, φs = π−2θs, φt = π−θt. (50)

The numbers of the three different types of building blocks filling out the space-time between t and t + 1 are N 14(t), N 41(t) and N 22(t). From this we compute thenumber of time-like links between t and t+1 as N TL

1 (t) = N 14(t)+N 41(t)+N 22(t)+2.The number of spatial links contained in the spatial slice at t is 2N 41(t), and each

spatial link at t belongs to two (4,1)-pyramids. Collecting all this information andusing the well-known expressions for Regge curvature in terms of dihedral angles[45], including boundary terms [46], the curvature contribution to the total actionis found to be

1

2

M

d3x√

g R(x) +

∂M

d2x√

h K (x) −→ (51)t

a

4π

β +

(2π − 4θt)

β + (2π − 4θs)

(N 41(t) + N 14(t) − N 22(t))

,

where the term in square brackets is always positive. The total volume contributing

to the cosmological term in the action is given by M

d3x√

g −→a3

β − 12

3

t

N 41(t)+ N 14(t)+

1

2N 22(t)

. (52)

17The constants α and β used in this appendix have nothing to do with the couplings α and β

appearing in the main text.

31



Putting in the appropriate (bare) coupling constants, this leads to a total discretizedaction

S [k, λ] = c0t − k

N 41 + N 14 − N 22

+ λ

N 41 + N 14 +

1

2N 22

, (53)

where now the total numbers of the different building blocks appear and where theconstants are given by

k =a

4πGN

− π

β + 2

β arccos

1

4β − 1+ arcsin

β − 1

2

β − 14

,

λ =a3Λ

24πGN

β − 1

2, c0 = −a

√β

2GN . (54)

In order to obtain the Lorentzian version of the action (53) (up to an imaginaryfactor −i), we need to replace β → −α such that l2

t = −αl2s , in accordance with

the notation in [15, 16]. To get the correct Lorentzian action, one then inverts theprescription given in [15, 16] by continuing the square-root expressions according to

√β → i√α and

β − 12 → i

α + 1

2 .

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j. ambjørn et al- lorentzian 3d gravity with wormholes via matrix models

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